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Topics in Algebraic Number Theory >> Content Detail



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Calendar

Lec #TOPICSKEY DATES
1Course Overview
2Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition)
3Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals
4Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, DiscriminantProblem set 1 due
5Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case
6Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, DiscriminantProblem set 2 due
7Discriminants and Ramification, Norms of Ideals
8Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case)Problem set 3 due
9Linearly Disjoint Extensions; Cyclotomic Fields (General Case)
10Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, LatticesProblem set 4 due
11Lattices and Ideal Classes, Minkowski's Theorem, Finiteness of the Class Group; Dirichlet's Units Theorem
12Proof of Dirichlet's Units TheoremProblem set 5 due
13Absolute Values; Completions of Fields with Respect to an Absolute Value, Examples; Dichotomy between Archimedean Nonarchimedean Absolute Values; Absolute Values Coming from Discrete Valuation Rings; Normalized Absolute Values (Places), Statement of the Product Formula for Number Fields; Classification of Completions of the Rational Numbers (Ostrowski's Theorem)
14In-class Midterm ExamProblem set 6 due
15Ostrowski's Theorem (cont.); Exponential and Logarithm Series; Hensel's Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values
16Extension of Nonarchimedean Absolute ValuesDetachable midterm exam due

Problem set 7 due
17Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified ExtensionsProblem set 8 due
18Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols
19Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic CaseProblem set 9 due
20More on Ray Class Groups; Idelic Interpretation
21Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet's Theorem and GeneralizationsProblem set 10 due
22Chebotarev Density Theorem; Arakelov Class Group
23Arakelov Class Group (cont.); Local Class Field Theory
24Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem
25Class Field Towers; Complex MultiplicationTake-home final exam due

 








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